Chapter 22 – Gauss’ Law and Flux• Lets start by reviewing some vector calculus

• Recall the divergence theorem

• It relates the “flux” of a vector function F thru a closed simply connected surface S bounding a region (interior volume) V to the volume integral of the divergence of the function F

• Divergence F => F

Volume integral of divergence of F = Surface (flux) integral of F

Mathematics vs Physics

• There is NO Physics in the previous “divergence theorem” known as Gauss’ Law

• It is purely mathematical and applies to ANY well behaved vector field F(x,y,z)

Some History – Important to know• First “discovered” by Joseph Louis Lagrange 1762

• Then independently by Carl Friedrich Gauss 1813

• Then by George Green 1825

• Then by Mikhail Vasilievich Ostrogradsky 1831

• It is known as Gauss’ Theorem, Green’s Theorem and Ostrogradsky’s Theorem

• In Physics it is known as Gauss’ “Law” in Electrostatics and in Gravity (both are inverse square “laws”)

• It is also related to conservation of mass flow in fluids, hydrodynamics and aerodynamics

• Can be written in integral or differential forms

Integral vs Differential Forms• Integral Form

• Differential Form (we have to add some Physics)

• Example - If we want mass to be conserved in fluid flow – ie mass is neither created nor destroyed but can be removed or added or compressed or decompressed then we get

• Conservation Laws

Continuity Equations – Conservation Laws

Conservation of mass in compressible fluid flow = fluid density, u = velocity vector

Conservation of an incompressible fluid = fluid density = constant here

Conservation of charge in electric current flowJ = current flux vector (amps/m2) , = charge density (coulombs/ m3)

Conservation of probability on Quantum Mechanics, j = probability flux vector, = probability

General Continuity Equation with source term s = source or sink – creation or annihilation

General Continuity Equation with s = 0

What are Continuity, Conservation Laws?The equation What it means

• Let F = V for a fluid then

• F = flux (mass flow) of fluid per unit area per unit of time (Kg/s-m2)

• If you integrate this over a closed surface (right hand side) you get the net mass change per unit time going INTO or OUT OF the surface

• This must be -m/t where m = mass inside the surface. Note minus sign –this depends on how we define the outward normal

• BUT m = dV and m/dt = /t dV

• Now equate the two sides of the equation

• We now get F = -/t or

• F +/t = 0 => Continuity equation

Gauss’ Law in Electromagnetism

• We start with an assumption about the E field from a point source.

• Assume it obeys Coulomb’s Law – ie inverse square law

Where er is a radial unit

vector away from the point

charge q

Compute the surface integral of E(r) over a sphere of radius r with

the charge q at the center. We will then use Gauss’ Law.

Surface integral over sphere

• Compute the surface integral of E(r) over a sphere of radius r with the charge q at thecenter.

• E(r) dA = 4r2 * kq/r2 = 4kq = q/0

• (NOTE: no r dependence) k=1/40

• E(r0) = 0 – this is true of ANY inverse square field (Gravity also)

• E(r=0) = (r) function ( at r=0, 0 otherwise)

What if we are not at the center of the sphere?We break the sphere into two imaginary regions – one sphere inside the

other but not centered.

Imagine there is only one charge in the smaller sphere and none between

The total flux when summing over both spheres is ZERO

Since E(r0) = 0 in between both spheres (no charges)

E dV = 0 = E dA (over both spheres)

But = E dA (total) = 0 = E dA (outer sphere) + = E dA (inner sphere)

Thus:

E dA (outer sphere) = - E dA (inner sphere)

But we know E dA (inner) = Q/0

Hence E dA (outer sphere) = Q/0 (not minus due to the way we oriented

the normal to the surface)

There was nothing special about the outer sphere, It could have been any

shape, Hence E dA = Q/0 where Q is the total charge enclosed.

More generally E = /0

Maxwell Eq #1 of 4 = charge density

Electric FluxDifferential flux

Integral flux

Flux is charge

enclosed Qs /0

This is Gauss’ Law

Coulomb’s Law from Gauss’ Law

• Assume we have a point charge at the center of a sphere and use Gauss’ Law

• And spherical symmetry

Hence we get Coulomb’s Law

Electrostatics – Here it is dipole moments

Sparks

Lightning

Human Lightning

A typical human has a capacitance of about 200-300 pf

Discharge can 10-20 KV+ and amps but microsecond long

Total energy is small so generally not harmful

Typical van de Graff generator is Positively charged but NOT always – depends on belt material

Flux =0 through sphere

Charged metal sphere – E=0 inside

Solving a spherical problem via Gauss’ LawAssume charge Q is spread uniformly over r<R

Gaussian Surface

Metal box in external E field“Faraday Cage”E=0 inside box

Using a “Gaussian Pillbox” and Gauss’ Law to solve for E field from a uniformly charged metal plate with charge per unit area = σ

Note E field is the same everywhere except inside metal (=0)

Two metal plates – a Capacitor

Charged ball and metal container

Charged ball inside neutral metal container

Charge moves to outside of metal container

Total flux = total charge enclosed/ε0

In this case it is ZERO

Lightning

• Approx 16 million lightning storms per year

• Speeds are very high – 60 Km/s (130,000 MPH)!!!

• Temperatures in bolts are very high – can be 30,000 C

• History of kite experiments – wet string = conductive = sparks fly (from key)

• Thomas-François Dalibard and De Lors May 1752

• Benjamin Franklin June 1752 (independent)

Some more on Lightning• Florida has the most US strikes

• Typ Negative Lightning bolt 30 Kilo amps, 5 Coulombs of charge and 500 Mega Joules of energy

• Large negative bolts can be 120 Kilo amps and 350 Coulombs of charge

• For reference – 1 Ton TNT ~ 4 Giga Joule of energy

• Typ Positive Lightning bolts are 10 times that of Negative Bolts

• Megawatts per meter of bolt are possible

• Typ PEAK power ~ 1 Tera watt (1000 nuclear power plants)

• Lightning heats air to 30,000 C or so and creates supersonic shock wave

• Lightning creates radio waves – these can clear particles from the Van Allen Belts (slots) and create low radiation zones

More Lightning Facts

• Norse mythology, Thor is the god of thunder

• Perkūnas - Baltic god of thunder

• Aztec had a god named Tlaloc

• Cyclops gave Zeus the Thunderbolt as a weapon

• Finnish mythology, Ukko (engl. Old Man) is the god of thunder

• Brescia, Italy in 1769 – lightning hit the Church of St. Nazaire, igniting

100 tons of gunpoweder that killed 3000 people

• Average lightning hit rate is 44+-5 Hz

• 1.4 billion flashes per year

• Lightning is dangerous - Georg Wilhelm Richmann July 1753 killed as he tried to repeat Ben Franklin’s experiment – hit in the head by a blue “ball of lightning”

• Venus, Jupiter and Saturn have lightning

More Lightning Trivia• Terawatt laser in NM induced minor lightning

• Rockets trailing wires can trigger lightning

• Elves (Emissions of Light and Very Low Frequency Perturbations from Electromagnetic Pulse Sources ) 250 miles up

• Lighning struck Apollo 12 after take off

• Triggered after above ground nuclear testing

• Triggered by volcanoes

• X-Ray, Gamma Ray 20 Mev and anti matter (positrons) seen from lightning

• Lightning strikes can induce ground magnetic “hot spots”

• Roy Sullivan held a Guinness World Record after surviving 7 different lightning strikes across 35 years.

• October 31 2005, sixty-eight dairy cows, died while taking shelter under a tress on a farm at Fernbrook Dorrigo, New South Wales

• December 8, 1963: Pan Am Flight 214 crashed 81 people were killed.

• November 2, 1994, lightning struck fuel tanks in Dronka, Egypt 469 fatalities

Global Lightning strike distribution

Lightning as Art

1902 Paris

Rimini Italy

Lightning is complex

Leader first then return strike

Ball Lightning – Japan

Volcano induced lightning

Shuttle hit – STS8 Challenger 1982/3

Light strike induced magnetism on the ground

Cloud to Cloud

Human Made Lightning – Tesla Coils

More Tesla Coil – Note Complex Streamers

Capacitance of simple systems

Type Capacitance Comment

Parallel-plate capacitorA: Aread: Distance

Coaxial cablea1: Inner radiusa2: Outer radiusl: Length

Pair of parallel wires[17]

a: Wire radiusd: Distance, d > 2al: Length of pair

Wire parallel to wall[17]

a: Wire radiusd: Distance, d > al: Wire length

Concentric spheresa1: Inner radiusa2: Outer radius

Two spheres,equal radius[18][19]

a: Radiusd: Distance, d > 2aD = d/2aγ: Euler's constant

Sphere in front of wall[18]

a: Radiusd: Distance, d > aD = d/a

Sphere a: Radius

Circular disc a: Radius

Thin straight wire,finite length[20][21][22]

a: Wire radiusl: LengthΛ: ln(l/a)